Question
Let $G=\left(V,E\right)$ be a directed graph such that all $v\in V$ have outdegree exactly 1. Let $\alpha \in \left[ 0,1\right)$. Denote by $P\left(v\right)$ the set of all paths ending at vertex $v$.
Does $\sum_{p\in P\left( v\right)}{\alpha^{\left|p\right|}}$ converge for all $D,\alpha$ and for any $v\in V$? If not, what can we say about $\alpha$s for which this series converges?
Observations so far
Firstly, $P\left(v\right)$ is infinite iff $v$ participates in a cycle in $G$, so it suffices to examine only such $v$'s, though I'm not sure if that helps.
To simplify the series, we can notice that $$\sum_{p\in P\left( v\right)}{\alpha^{\left|p\right|}} = \sum_{n=1}^{\infty}{\left| P\left(v\right)\cap V^n \right|\alpha^n}$$ so if $\left| P\left(v\right)\cap V^n \right|$ grows slow enough (or is even bounded) the series will indeed converge, or in other words we wish to prove some notion of sparsity of $P\left(v\right)$ in $V^\ast$. For this, we need to make some combinatorial observation about the number of possible paths given that the outdegree is exactly one, which is where I am stuck (if the claim is indeed true).
Motivation
I am reading a paper about voting in social networks and am trying to determine whether the voting score in a transitive proxy voting system with exponential damping is necessarily finite, as defined in section 3.1. This is not explicitly stated in the paper.